FAST NUMERICAL CONFORMAL MAPPING OF BOUNDED...

FAST NUMERICAL CONFORMAL MAPPING OF BOUNDED MULTIPLYCONNECTED REGIONS VIA INTEGRAL EQUATIONS

LEE KHIY WEI

UNIVERSITI TEKNOLOGI MALAYSIA

FAST NUMERICAL CONFORMAL MAPPING OF BOUNDED MULTIPLY

CONNECTED REGIONS VIA INTEGRAL EQUATIONS

LEE KHIY WEI

A thesis submitted in fulfilment of the

requirements for the award of the degree of

Master of Philosophy

Faculty of Science

Universiti Teknologi Malaysia

DECEMBER 2016

iii

To my beloved family, lecturers and my friends, thank you for your great support in

term of physically and mentally.

iv

ACKNOWLEDGEMENT

First of all, I wish to express my deepest gratitude to my supervisor, Assoc.

Prof. Dr. Ali Hassan Mohamed Murid, who has supervised the research. I thank him

for his enduring patience toward the research. Moreover, I wish to express my gratitude

to my co-supervisor Assoc. Prof. Dr. Ali Wahab Kareem Sangawi from Department

of Computer, College of Basic Education, Charmo University, Iraq, for his guidance

for the knowledge of integral equation method for numerical conformal mapping. My

special thanks are also due to Prof. Dr. Mohamed M.S. Nasser, from Qatar University,

Qatar, for contributing ideas, guiding in Matlab programming, and collaboration in

some parts of my research.

I would also like to express my gratitude to my family members, friends and

UTM staffs who have helped me directly or indirectly to complete my research. They

had been very kind and had tried their best in providing assistance.

I further wish to express my indebtedness to the Malaysian Ministry of Higher

Education (MOHE) for financial support through the Research Management Centre

(RMC), Universiti Teknologi Malaysia (FRGS R.J130000.7809.4F637).

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ABSTRACT

This study presents a fast numerical conformal mapping of bounded multiply

connected region onto a disk with circular slits, an annulus with circular slits, circular

slits, parallel slits and radial slits regions and their inverses using integral equations

with Neumann type kernel and adjoint generalized Neumann kernel. A graphical user

interface is created to illustrate the effectiveness of the approach for computing the

conformal maps of bounded multiply connected regions and image transformations

via conformal mappings. Some image transformation results are shown via graphical

user interface. This study also presents a fast numerical conformal mapping of

bounded multiply connected region onto second, third and fourth categories of Koebe’s

canonical slits regions using integral equations with adjoint generalized Neumann

kernel. The integral equations are discretized using Nystrom method with trapezoidal

rule. For regions with corners, the integral equations are discretized using Kress’s

graded mesh quadrature. All the linear systems that arised are solved using generalized

minimal residual method (GMRES) or least square iterative method powered by fast

multipole method (FMM). The interior values of the mapping functions and their

inverses are determined by using Cauchy integral formula. Some numerical examples

are presented to illustrate the effectiveness for computing the conformal maps of

bounded multiply connected regions. This study also discussed a fast numerical

conformal mapping of bounded multiply connected regions onto fifth category of

Koebe’s canonical regions using integral equations with the generalized Neumann

kernel. An application of fast numerical conformal mapping to some coastal domains

with many obstacles is also shown.

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ABSTRAK

Kajian ini mempersembahkan satu pemetaan konformal berangka bagi rantau

terkait berganda terbatas yang pantas ke cakera dengan belahan membulat, annulus

dengan belahan membulat, rantau belahan membulat, rantau belahan jejari dan rantau

belahan selari dan songsangannya menggunakan persamaan kamiran dengan inti

jenis Neumann dan inti Neumann dampingan terilak. Satu antara muka pengguna

bergrafik dicipta untuk menunjukkan keberkesanan kaedah dalam pengiraan permetaan

konformal berangka rantau terkait berganda terbatas dan transformasi imej melalui

pemetaan konformal. Beberapa hasil transformasi imej telah ditunjukkan melalui

antara muka pengguna bergrafik. Kajian ini juga mempersembahkan satu pemetaan

konformal berangka rantau terkait berganda terbatas yang pantas ke kategori kedua,

ketiga dan keempat rantau berkanun Koebe menggunakan persamaan kamiran bersama

inti Neumann terilak. Persamaan kamiran tersebut didiskretkan menggunakan kaedah

Nystrom dengan petua trapezoid. Bagi rantau berpenjuru, persamaan kamiran tersebut

telah didiskretkan menggunakan kuadratur Kress bergred. Semua sistem linear yang

terhasil diselesaikan menggunakan kaedah reja minimum teritlak (GMRES) atau

kaedah lelaran kuasa dua terkecil yang dikuasai oleh kaedah multikutub pantas (FMM).

Nilai-nilai pedalaman bagi fungsi pemetaan dan pemetaan sonsangannya dikenal

pasti dengan menggunakan formula kamiran Cauchy. Beberapa contoh berangka

dipersembahkan bagi menggambarkan keberkesanan pengiraan permetaan konformal

rantau terkait berganda terbatas. Kajian ini juga membincang pemetaan konformal

berangka rantau terkait berganda terbatas yang pantas ke kategori kelima rantau

berkanun Koebe menggunakan persamaan kamiran bersama inti Neumann terilak.

Aplikasi pemetaan konformal berangka kepada domain pantai dengan banyak halangan

juga ditunjukkan.

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TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES xi

LIST OF FIGURES xii

LIST OF APPENDICES xvi

1 INTRODUCTION 1

1.1 Introduction 1

1.2 Research Background 2

1.3 Research Statements 3

1.4 Research Objectives 4

1.5 Scope of the Study 4

1.6 Significance of the Study 5

1.7 Organization of the Thesis 5

2 LITERATURE REVIEW 7

2.1 Introduction 7

2.2 Conformal Mapping 7

2.2.1 Conformal Mapping of Multiply

Connected Region 8

2.3 The Numerical Methods for Solving Integral Equations 13

2.3.1 The Nystrom Method with Trapezoidal Rule 13

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2.3.2 The Fast Multipole Method (FMM) 14

2.3.3 Iterative Methods for solving Linear System 15

2.3.4 Kress’s Graded Mesh Quadrature 19

2.4 Potential Flows in Multiply Connected Coastal Domain 20

3 NUMERICAL CONFORMAL MAPPING OF

BOUNDED MULTIPLY CONNECTED REGIONS

ONTO FIRST CATEGORY OF KOEBE’S

CANONICAL SLITS REGIONS 24

3.1 Boundary Integral Equations for Conformal

Mappings onto First Category of Koebe’s

Canonical Slits Regions 25

3.1.1 Integral Equations for Computing

Conformal Mapping onto a Disk with

Circular Slits Region 25


Conformal Mapping onto Annulus

with Circular Slits Region 27


Conformal Mapping onto Circular

Slits Region 29


Conformal Mapping onto Parallel Slits Region 30


Conformal Mapping onto Radial Slits

Region 32

3.2 Numerical implementation 34

3.2.1 Solving Integral Equation for Conformal

Mapping onto Unit Disk with Circular Slits 35


Mapping onto Annulus with Circular Slits 38

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Mapping onto Circular Slits 42


Mapping onto Parallel Slits 47


Mapping onto Radial Slits 52

3.3 Numerical Examples 57

3.3.1 Graphical User Interface 66

4 NUMERICAL CONFORMAL MAPPING OF

BOUNDED MULTIPLY CONNECTED REGIONS

ONTO SECOND, THIRD AND FOURTH

CATEGORY OF KOEBE’S CANONICAL SLITS

REGIONS 72

4.1 Boundary Integral Equations for Conformal

Mappings onto second, third and fourth Koebe’s

canonical slits regions 73

4.1.1 Disk with logarithmic spiral slits map 73

4.1.2 Annulus with logarithmic spiral slits map 76

4.1.3 Logarithmic Spiral Slits Map 78

4.1.4 Straight Slits Map 80

4.2 Numerical Implementation 82

4.3 Numerical Boundary Integral Equation for Non-

smooth Region 89


5 CONFORMAL MAPPING APPROACH

FOR POTENTIAL FLOWS IN MULTIPLY

CONNECTED COASTAL DOMAINS 104

5.1 Uniform Flow 106

5.2 Single Vortex or Multiple Vortices 110

5.3 Flow Induced by a Source (or a Sink) on the

Boundary of Coast Line 111

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5.4 Flow Induced by a pair of Source and Sink 113


6 SUMMARY AND SUGGESTIONS FOR FUTURE

RESEARCH 120

6.1 Summary of the Research 120

6.2 Suggestions for Future Research 123

REFERENCES 124

Appendices A-E 132-207

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LIST OF TABLES

TABLE NO. TITLE PAGE

3.1 Error Norm for Example 3.1. 57

3.2 The values of constants aj , bj , ξj and σj in Example 3.2. 63

4.1 The values of constants aj , bj , ξj and σj in Example 4.1. 92

4.2 Radii comparison for Example 4.1 with Nasser’s

method 100

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LIST OF FIGURES

FIGURE NO. TITLE PAGE

2.1 Multiply connected region onto first category of

Koebe’s canonical slits regions. 10

2.2 Mapping of a bounded multiply connected region Ω

onto second, third and fourth categories of Koebe’s

canonical slits regions. 11

2.3 (a) Coastal domain around Mersing islands and the

west coast of east Malaysia. (b) Coastal domain

around the Japan islands and the coastline of East Asia. 22

3.1 The original region Ω and its images for Example 3.1. 58

3.2 The inverse image of Ω1 for Example 3.1. 59

3.3 The inverse image of Ω2 region for Example 3.1. 59




3.7 Time taken of the numerical experiment varies

number of nodes for computing conformal mapping

onto disk with circular slits regions Ω1. 60



onto annulus with circular slits regions Ω2. 61



onto circular slits regions Ω3. 61

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onto parallel slits regions Ω4. 62



onto radial slits regions Ω5. 62







3.18 GUI for conformal mapping onto disk with circular

slits and annulus with circular slits with fixed

boundaries selected. 68

3.19 GUI for conformal mapping onto circular slits,

parallel slits and radial slits with fixed boundaries selected. 69

3.20 GUI for conformal mapping onto disk with circular

slits and annulus with circular slits with boundaries

generated using growcut algorithm. 70

3.21 GUI for conformal mapping onto circular slits,

parallel slits and radial slits with boundaries

generated using growcut algorithm. 71


4.2 The inverse image of Ua for Example 4.1. 94

4.3 The inverse image of Ub region for Example 4.1. 94

4.4 The inverse image of Uc region for Example 4.1. 95

4.5 The inverse image of Ud region for Example 4.1. 95

4.6 The images for Example 4.1. 96

4.7 The inverse image of Ua for Example 4.1. 97

4.8 The inverse image of Ub region for Example 4.1. 97

4.9 The inverse image of Uc region for Example 4.1. 98

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4.10 The inverse image of Ud region for Example 4.1. 98

4.11 Computing time versus number of nodes for Example 4.1. 99

4.12 Memory allocation versus number of nodes for

Example 4.1. 99






5.1 (a) Coastal domain around Mersing Islands and the

west coast of east Malaysia. (b) Coastal domain

around the Japan islands and the coastline of East Asia. 105

5.2 A coastal domain ω in the z-plane. 106

5.3 An unbounded multiply connected domain Λ in the

ζ-plane. 107

5.4 A bounded multiply connected domain D in the ξ-plane. 108

5.5 A bounded multiply connected domain S in the η-plane. 108

5.6 The canonical multiply connected region Ωu in w-plane. 109

5.7 The canonical multiply connected region Ωb in w-plane. 111

5.8 The canonical multiply connected region ΩS in w-plane. 114

5.9 (a) Original region from Figure 5.1(a). (b) Region Λ

mapped from original region. (c) Region D mapped

from region Λ. (d) Region S mapped from region D. 115

5.10 (a) Region ΩU mapped from region S. (b) Region

ΩB mapped from region S. (c) Region ΩS mapped

from region D. (d) Region ΩS mapped from region D. 116

5.11 Potential flows in a coastal domain with three islands.

(a) Streamlines of uniform flow. (b) Streamlines of a

single vortex. (c) Streamlines of a source point on the

wall boundary. (d) Streamlines of source-sink pair.

(e) Streamlines of multiple vortices. (f) Streamlines

of a uniform flows with four points vortices. 118

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5.12 Potential flows in a coastal domain of with three

Japan islands. (a) Streamlines of uniform flow. (b)

Streamlines of a single vortex. (c) Streamlines of a

source point on the wall boundary. (d) Streamlines of

source-sink pair. (e) Streamlines of multiple vortices.

(f) Streamlines of a uniform flows with four points

vortices. 119

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LIST OF APPENDICES

APPENDIX TITLE PAGE

A Papers Published During The Author’s Candidature 132 - 133

B Derivation of Integral Equations for Conformal

Mapping of Bounded Multiply Connected Regions

onto Disk with Circular Slits, Annulus with

Circular Slits, Circular Slits and Parallel Slits Regions 134 - 144

C Derivation of Integral Equations for Conformal

Mapping of Bounded Multiply Connected Regions

onto Disk with Logarithmic Spiral Slits Regions 145 - 148

D Sample of Computer Programs Coded in Matlab

R2013a for Example 3.1 149 - 189

E Sample of Computer Programs Coded in Matlab

R2013a for Example 4.1 190 - 207

CHAPTER 1

INTRODUCTION

1.1 Introduction

Conformal mapping is a one-to-one and onto function of a complex variable.

It transforms a planar region to another planar region. The magnitudes as well as the

angles between two curves are preserved under conformal mapping. As complicated

regions can be transformed into simpler regions, various problems in the engineering

and science fields can be solved easily.

Conformal mapping of multiply connected regions onto five categories with

39 canonical slits regions had been introduced by Koebe (Koebe, 1916). Several

numerical methods have been done to compute the conformal mapping of multiply

connected regions onto canonical slits regions. A method has been presented to

compute the conformal mapping onto all of 39 canonical slits regions in a unified ways

based on uniquely solvable boundary integral equations with generalized Neumann

kernel (Nasser, 2009a; Nasser, 2009b; Nasser, 2011; Nasser, 2013). Several other

works have also been presented to compute conformal mapping of bounded and

unbounded multiply connected regions onto several canonical slits regions using

integral equation method (Amano, 1998; Okano et al., 2003; Yunus et al., 2012; Yunus

et al., 2014a; Yunus et al., 2013; Yunus et al., 2014b; Sangawi et al., 2012a; Sangawi

et al., 2012b; Sangawi et al., 2011; Sangawi et al., 2013; Sangawi, 2014a; Sangawi,

2014b; Sangawi and Murid, 2013).

2

The Nystrom method with the trapezoidal rule has been used effectively to

discretize the integral equations since the functions in the integral equations are all

periodic (Davis and Rabinowitz, 1984). Normally the linear system obtained from

discretization can be solved using the Gauss elimination method. However, the

solution of the system is difficult to obtain when the number of nodes for discretization

is large. A fast approach named as fast multipole method (FMM) has been proposed

by Greengard and Rokhlin (1987) for faster particle simulations in plasma physics.

Nasser and Al-Shihri (2013) proposed a fast method to solve integral equations with

generalized Neumann kernel and compute conformal mapping using fast multipole

method (FMM) and restarted version of generalized minimal residual (GMRES)

method. This approach is fast, accurate and able to obtain the solution with very high

number of nodes. However, the system obtained must be linear in order to be solved

using GMRES method and also, this approach cannot be applied to overdetermined

system.

1.2 Research Background

New integral equations have been derived to compute conformal mapping of

bounded multiply connected regions onto first category of Koebe’s canonical slits

regions, namely disk with circular slits, annulus with circular slits, circular slits,

parallel slits and radial slits regions. However, the approach requires three integral

equations for each canonical region.

For some of these new integral equations, the systems obtained after

discretization are overdetermined systems. An overdetermined system can been solved

using QR decomposition and least square method (Hayami et al., 2010). But classical

method like QR decomposition take high cost in time and memory, especially for large

number of nodes.

Nasser and Al-Shihri (2013) and Nasser (2015) proposed a fast method to

solve integral equations with generalized Neumann kernel and compute conformal

3

mapping using fast multipole method (FMM) and restarted version of generalized

minimal residual (GMRES) method. This approach can be applied to these new

integral equations for numerical conformal mapping.

Integral equation methods are not only useful for computing conformal

mapping of region with smooth boundaries, they are also applicable to compute

conformal mapping of regions with non-smooth boundaries. Yunus (2013) had

done numerical experiment for region with corners using Kress quadrature rule

(Kress, 1990).

The integral equations derived in Sangawi (2014a), Sangawi (2014b), Sangawi

and Murid (2013) and Sangawi et al. (2013) are solved numerically using MATLAB

solver that requires explicitly defined coefficient matrix. The operation for this

approach are O((m+ 1)2n2), where m+ 1 is the number of connectivity of the region

and n is number of node for each boundary. It is not suitable for higher connectivity and

more challenging region problems. Some improvements need to be done for reducing

the operations and memory requirement for numerical conformal mapping in order to

apply in real problem.

1.3 Research Statements

The research problem is to apply fast multipole method and LSQR method for

computing conformal mapping of bounded multiply connected regions with smooth

boundaries onto first category canonical regions including disk with circular slits

region, annulus with circular slits region, circular slits region, parallel slits region

and radial slits region using new integral equations developed by Ali Wahab Kareem

Sangawi in Appendix B and Sangawi et al. (2013). This research problem is also to

apply fast multipole method and generalized minimal residual (GMRES) method for

computing conformal mapping of bounded multiply connected regions with smooth

and non-smooth boundaries onto second, third and fourth categories of canonical

regions developed by Ali Wahab Kareem Sangawi in Appendix C and Sangawi

(2014a), Sangawi (2014b) and Sangawi and Murid (2013).

4

1.4 Research Objectives

The objectives of this research are:

(i) To construct fast numerical technique to compute conformal

mapping of bounded multiply connected regions with smooth

boundaries onto first category of Koebe’s canonical slits regions

based on new integral equations developed by Ali Wahab Kareem

Sangawi in Appendix B and Sangawi et al. (2013) by using LSQR

and fast multipole method and illustrate the results by constructed a

graphical user interface.

(ii) To construct fast numerical technique to compute conformal

mapping of bounded multiply connected region with smooth and

non-smooth boundaries onto second, third and fourth categories

of canonical regions based on new integral equations developed

by Ali Wahab Kareem Sangawi in Appendix C and Sangawi

(2014a), Sangawi (2014b) and Sangawi and Murid (2013) based on

GMRES method, FMM, and Kress quadrature rule.

(iii) To develop graphical user interface to illustrate the image

transformation based on objective (i).

(iv) To illustrate an application of fast numerical conformal mapping to a

coastal domain with many obstacles.

1.5 Scope of the Study

This research focuses on boundary integral equation methods to compute the

conformal mapping function of bounded multiply connected region onto canonical

slits regions. The fast multipole method is considered to compute matrix-vector

multiplication and the iterative methods including GMRES method and LSQR method

are considered to solve the linear system. Normally the integral equation will

be discretized using Nystrom’s method with trapezoidal rule when the function is

periodic. Kress quadrature rule is used to discretize the integral equation when

conformal mappings of non-smooth regions are computed.

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1.6 Significance of the Study

Conformal mapping is important in solving several problems that arises from

science, engineering and medical fields. Many real problems involve smooth and non-

smooth regions. Conformal mapping play the role in transforming the complicated

region to a standard region, where the problem can be solved easily. The solution is

obtained after transforming back to the original region.

The usual algorithm without fast multipole method takes a longer time and

requires much memories in the computation by using computer. When fast multipole

method is applied into the computation, the time taken to compute shorten from

O(n2(m + 1)2) to O(n(m + 1)) with less memory needed. The fast algorithm is

capable to compute the mapping of more complex regions. Many hospitals have

also been using medical imaging extensively for rapid diagnosis with visualization

and quantitative assessment. Conformal mapping algorithm can be applied to medical

image processing, which help the medical experts in diagnosis (Gu et al., 2004).

1.7 Organization of the Thesis

The thesis is organized into five chapters. Chapter 1 contains six sections

which are introduction, problem statements, objectives of the study, scope of the study,

significance of the study and organization of the thesis.

Chapter 2 begins with some explanations on concepts of conformal mapping

of bounded multiply connected regions onto some Koebe’s canonical regions followed

by auxiliary materials. Furthermore, Cauchy’s integral formula, FMM and Iterative

Methods for solving linear system are also briefly discussed.

Chapter 3 discusses integral equations developed in Sangawi et al. (2013)

for computing conformal mapping of bounded multiply connected regions onto first

category of Koebe’s canonical slits regions. Chapter 3 also presents the numerical

6

computation for the conformal mapping of any bounded multiply connected region

with smooth boundaries onto first category of canonical regions using FMM, GMRES

and LSQR. Some numerical examples including graphical user interface are also given.

Chapter 4 discusses new integral equations for computing conformal mapping

of bounded multiply connected regions onto second, third and fourth Koebe’s

canonical slits regions. The numerical computation for the conformal mapping of

any bounded multiply connected region with smooth and non-smooth boundaries onto

second, third and fourth Koebe’s canonical slits regions using FMM and GMRES

method is also discussed in Chapter 4. Some numerical examples are also given.

An example on application of conformal mapping to compute potential flow

with a coastal domain with many obstacles is given in Chapter 5. It is shown how to

construct uniform flow, uniform flow with a point vortex, multiple vortices, uniform

flow with source at boundary with a coastal domain by means of conformal mapping.

The concluding chapter, Chapter 6 contains summary of the research and some

recommendations for future research.

124

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